# Twisted L-Functions and Monodromy. (AM-150)

Nicholas M. Katz
Pages: 237
https://www.jstor.org/stable/j.ctt7t2qp

1. Front Matter
(pp. i-iv)
(pp. v-2)
3. Introduction
(pp. 3-22)

The present work grew out of an entirely unsuccessful attempt to answer some basic questions about elliptic curves overQ. Start with an elliptic curve E overQ, say given by a Weierstrass equation

${\text{E:}}\quad \;{{\text{y}}^2} = 4{{\text{x}}^3} - {\text{ax}} - {\text{b}}$,

with a, b integers and${{\text{a}}^3} - 27{{\text{b}}^2} \ne \;0$. By Mordell’s theorem [Mor], the group${\text{E(Q)}}$ofQ–rational points is a finitely generated abelian group. The dimension of theQ–vector space${\text{E(Q)}} \otimes {{\text{Z}}^{\text{Q}}}$is called the Mordell–Weil rank, or simply the rank, of E. Thus we get a function

$\{ ({\rm{a}},{\rm{b}}){\rm{in}}{{\rm{Z}}^2}{\rm{with}}{{\rm{a}}^3} - 27{{\rm{b}}^2} \ne 0\} {\rm{ }} \to {\rm{ }}\{ {\rm{nonnegativeintegers}}\}$

defined by

$({\text{a,}}\;{\text{b)}} \mapsto {\text{the rank of the curve}}\;{{\text{y}}^2} = 4{{\text{x}}^3} - {\text{ax}} - {\text{b}}$.

4. Part I: Background Material
• Chapter 1: “Abstract” Theorems of Big Monodromy
(pp. 23-42)

(1.0.1) It will be convenient to introduce two generalizations of the notion of pseudoreflection. Suppose we are given a finite–dimensional vector space V over a field K. We write GL(V) for${\text{Au}}{{\text{t}}_{\text{K}}}({\text{V)}}$, so long as there is no ambiguity about the field K. Recall that an element A in GL(V) is called a pseudoreflection if its space of fixed points, Ker(A–1), has codimension one in V, or, equivalently, if the quotient spaceV/Ker(A–1) has dimension one.

(1.0.2) Given an integer${\text{r}} \geqslant \;{\text{0}}$, and an element A in GL(V), we say that A has drop r if Ker(A–1) has...

• Appendix to Chapter 1: A Result of Zalesskii
(pp. 43-50)
• Chapter 2: Lefschetz Pencils, Especially on Curves
(pp. 51-70)

(2.0.1) We work over an algebraically closed field k. Let X/k be a proper smooth connected kscheme of dimension${\text{n}} \geqslant {\text{1}}$, andLon X a very ample invertible${O_X}{\kern 1pt} - {\kern 1pt} {\text{module}}$-module. We embed X in${\text{P(}}{{\text{H}}^0}({\text{X,}}\;{\text{L)}}$, the projective space of hyperplanes in${{\text{H}}^0}({\text{X,}}\;{\text{L)}}$, in the usual way: x in X(k) is mapped to the hyperplane in${{\text{H}}^0}({\text{X,}}\;{\text{L)}}$consisting of those global sections ofLwhich vanish at x. Equivalently, we give ourselves X as a closed subscheme of a projective space${\text{p}}$in such a way that both the following conditions are satisfied:

(2.0.1.1)$\mathcal{L}\;{\text{is}}\;{O_X}(1): = \;{\text{the pullback to X}}\;{\text{of}}\;{O_{\text{P}}}(1)$,

(2.0.1.2) the restriction map induces an isomorphism...

• Chapter 3: Induction
(pp. 71-78)

(3.0.1) Let G be a group,${\text{H}} \subset {\text{G}}$a subgroup, R a commutative ring, and V a left R[H]–module. There are two standard notions of the induction of V from H to G. The first, which we call “standard” induction, is

(3.0.1.1)${\text{In}}{{\text{d}}_{\text{H}}}^{\text{G}}({\text{V)}}\;{\text{:}} = \;{\text{R[G]}}{ \otimes _{{\text{R[H]}}}}{\kern 1pt} {\text{V}}$

with its structure of left R[G]–module through the first factor.

The second, which we call Mackey induction, is

(3.0.1.2)${\text{MaIn}}{{\text{d}}_{\text{H}}}^{\text{G}}({\text{V)}}\;{\text{:}} = \;{\text{Ho}}{{\text{m}}_{{\text{left}}\;{\text{R[H]}} - {\text{mod}}}}({\text{R[G],}}\;{\text{V)}} = \;{\text{Ho}}{{\text{m}}_{{\text{left}}\;{\text{H}} - {\text{sets}}}}({\text{G,}}\;{\text{V)}}$,

which becomes a left R[G]-module by defining

$({{\text{L}}_{\text{g}}}\varphi ){\text{(x)}}: = \;\varphi ({\text{xg)}}$.

(3.0.2) For standard induction, we get, for any left R[G]-module W, one version of Frobenius reciprocity:

(3.0.2.1)${\text{Ho}}{{\text{m}}_{{\text{left}}\;{\text{R[H]}} - {\text{mod}}}}({\text{V,}}\;{\text{W|H)}} \cong \;{\text{Ho}}{{\text{m}}_{{\text{left}}\;{\text{R[G]}} - {\text{mod}}}}({\text{In}}{{\text{d}}_{\text{H}}}^{\text{G}}({\text{V),}}\;{\text{W)}}$,

the isomorphism being$\psi \mapsto \;({\text{the}}\;{\text{map}}\;{{\text{g}}^ \otimes }{\text{v}} \mapsto {\text{g}}\psi {\text{(v)}}$. Taking for W the trivial...

• Chapter 4: Middle Convolution
(pp. 79-84)

(4.0.1) We fix a prime number$\ell$. We work on${{\text{A}}^1}$over an algebraically closed field k in which$\ell$is invertible. We wish to define a certain class${P_{{\text{conv}}}}$of irreducible middle extension${{{\text{\bar Q}}}_\ell }\, - \,{\text{sheaves}}\;F$on${{\text{A}}^1}$. Given an irreducible middle extension${{{\text{\bar Q}}}_\ell }\, - \,{\text{sheaf}}\;F$on${{\text{A}}^1}$(or, equivalently, a nonpunctual irreducible perverse sheaf${\text{K}} = F[1]\;{\text{on}}\;{{\text{A}}^1}$), denote by

(4.0.1.1)${\text{S:}}\, = \;{\text{Sing(}}F{)_{{\text{finite}}}}$

finite the finite set of points in${{\text{A}}^1}$at which$F$is not lisse.

(4.0.2) We say that$F$lies in${P_{{\text{conv}}}}$if

(4.0.2.1)${\text{rank(}}F)\; + \;\# {\text{S}}\; + \;{\Sigma _{{\text{t}}\;{\text{in}}\;{\text{S}} \cup {\text{\{ }}\infty {\text{\} }}}}\;{\text{Swa}}{{\text{n}}_{\text{t}}}(F)\; \geqslant \;3$.

(4.0.3) If k has characteristic zero, then among all irreducible middle extensions$F$, only the constant sheaf${{{\text{\bar Q}}}_\ell }\,$...

5. Part II: Twist Sheaves, over an Algebraically Closed Field
• Chapter 5: Twist Sheaves and Their Monodromy
(pp. 85-116)

(5.0.1) In this section, we make explicit the “families of twists” we will be concerned with. We fix an algebraically closed field k, a proper smooth connected curve C/k whose genus is denoted g, and a prime number$\ell$invertible in k. We also fix an integer${\text{r}} \geqslant {\text{1}}$, and an irreducible middle extension${{{\text{\bar Q}}}_\ell }\, - \,{\text{sheaf}}\;F$on C of generic rank r. This means that for some dense open set U in C, with${\text{j}}\;{\text{:}}\;{\text{U}} \to {\text{C}}$the inclusion,$F\,|\,U$is a lisse sheaf of rank r on U which is irreducible in the sense that the corresponding r–dimensional${{{\text{\bar Q}}}_\ell }$–representation of${\pi _1}({\text{U)}}$...

6. Part III: Twist Sheaves, over a Finite Field
• Chapter 6: Dependence on Parameters
(pp. 117-124)

(6.0.1). The following lemma is standard. We include it for ease of reference.

Lemma 6. 0. 2 Let T be an arbitrary scheme, X/T a proper smooth T–scheme with geometrically connected fibres everywhere of dimension N,Lan invertible${O_{\text{X}}}$–module, and L in${{\text{H}}^0}({\text{X,}}\;L)$a global section. Suppose L is nonzero on each geometric fibre of X/T, i. e., for every geometric point t of T, the image${{\text{L}}_{\text{t}}}$of L in${{\text{H}}^0}({{\text{X}}_{\text{t}}},\;{L_{\text{t}}})$is nonzero. Then the locus “L = 0 as section ofL”, call it Z, is a Cartier divisor in X, which is flat over...

• Chapter 7: Diophantine Applications over a Finite Field
(pp. 125-146)

(7.0.1) In this section, we work over a finite field k, of cardinality q and characteristic p. We fix a proper, smooth, geometrically connected curve C/k of genus g, an effective divisor D on C of degree${\text{d}} \geqslant {\text{2g}} + 1$, a prime number$\ell$invertible in k, an integer${\text{r}} \geqslant {\text{1}}$, and a geometrically irreducible middle extension${{{\text{\bar Q}}}_\ell }\, - \,{\text{sheaf}}\;F$on C of generic rank r. We denote by${\text{Sing(}}F)\; \subset \;{\text{C}}$the finite set of closed points of C at whichFis not lisse, and by${\text{Sing(}}F{)_{{\text{finite}}}}$the intersection${\text{Sing(}}F)\; \cap ({\text{C}}\, - \,{\text{D)}}$.

The space

(7.0.1.1)${\text{X}}\;{\text{:}} = \;Fct({\text{C,}}\;{\text{d,}}\;{\text{D,}}\;{\text{Sing(}}F{)_{{\text{finite}}}})$

has a natural structure of scheme over k, cf. Proposition 6.1.10....

• Chapter 8: Average Order of Zero in Twist Families
(pp. 147-178)

(8.0.1) In this section, we work over a finite field k of odd characteristic. We give ourselves data${\text{(C/k, D, }}\ell {\text{, r, }}F{\text{, }}\chi {\text{, }}\iota {\text{, w)}}$as in 7.0. We suppose that after extension of scalars from k to${{\text{\bar k}}}$, our data${\text{(C/k, D, }}\ell {\text{, r, }}F{\text{, }}\chi {\text{)}}$satisfies all the hypotheses of Theorem 5.5.1.

(8.0.2) We further suppose that$F({\text{w}}/2)$is symplectically self–dual on C/k, and that$\chi$has order 2.

Then, by Poincaré duality,$G(({\text{w}} + 1)\,/\,2)$is orthogonally self–dual as a lisse sheaf on

${\text{X}}\;{\text{:}}\, = Fct({\text{C,}}\;{\text{d,}}\;{\text{D,}}\;{\text{Sing(}}F{)_{{\text{finite}}}})$.

By Theorem 5.5.1,$G$has${{\text{G}}_{{\text{geom}}}}$either SO(N) or O(N).

(8.1.1) Given a finite extension E/k, and f in X(E), we define the analytic...

7. Part IV: Twist Sheaves, over Schemes of Finite Type over $\mathbb{Z}$
• Chapter 9: Twisting by “Primes”, and Working over $\mathbb{Z}$
(pp. 179-206)

(9.0.1) In this section, we work over an arbitrary scheme T, which will play the role of a parameter scheme in what follows. We fix a proper, smooth, geometrically connected curve C/T of genus g, and an integer${\text{d}} \geqslant {\text{2g}} + 1$. We denote by${\text{Ja}}{{\text{c}}^{\text{d}}}({\text{C/T)}}$, or simply${\text{Ja}}{{\text{c}}^{\text{d}}}$, the open and closed subscheme of${\text{Pi}}{{\text{c}}_{{\text{C/T}}}}$formed by divisor classes of degree d. We denote by${\text{Di}}{{\text{v}}^{\text{d}}}({\text{C/T)}}$the space of effective divisors in C of degree d. Thus for any T–scheme Y, a Y–valued point of${\text{Di}}{{\text{v}}^{\text{d}}}({\text{C/T)}}$is a closed subscheme of${\text{C}}{ \times _{\text{T}}}\,{\text{Y}}$which is finite and locally free over Y...

• Chapter 10: Horizontal Results
(pp. 207-234)

(10.0.1) We fix a prime number$\ell$, an integer${\text{n}} \geqslant {\text{2}}$, and a character

$\chi :\;{\mu _{\text{n}}}({\text{Z[1/}}\ell {\text{n,}}\;{\zeta _{\text{n}}}{\text{])}}\; \to \;{({{{\text{\bar Q}}}_\ell })^ \times }$

of order n. We fix a nonempty connected normal${\text{Z[1/}}\ell {\text{n,}}\;{\zeta _{\text{n}}}]\,$–scheme T of finite type. We fix a proper, smooth, geometrically connected curve C/T of genus g. We suppose given an effective Cartier divisor S in C which is finite etale over T of degree${\text{s}} \geqslant {\text{0}}$(with the convention that S is empty if e = 0). We suppose given a lisse${{{\text{\bar Q}}}_\ell }\, - \,{\text{sheaf}}\;F$on C – S of rank${\text{r}} \geqslant {\text{1}}$. If n is 4 or 6, we suppose that${\text{r}} \leqslant 2$. We suppose given an integer w, and...

8. References
(pp. 235-240)
9. Index
(pp. 241-249)